Metamath Proof Explorer


Theorem cnvuni

Description: The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004)

Ref Expression
Assertion cnvuni 𝐴 = 𝑥𝐴 𝑥

Proof

Step Hyp Ref Expression
1 elcnv2 ( 𝑦 𝐴 ↔ ∃ 𝑧𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝐴 ) )
2 eluni2 ( ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝐴 ↔ ∃ 𝑥𝐴𝑤 , 𝑧 ⟩ ∈ 𝑥 )
3 2 anbi2i ( ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝐴 ) ↔ ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ∃ 𝑥𝐴𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
4 r19.42v ( ∃ 𝑥𝐴 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) ↔ ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ∃ 𝑥𝐴𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
5 3 4 bitr4i ( ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝐴 ) ↔ ∃ 𝑥𝐴 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
6 5 2exbii ( ∃ 𝑧𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝐴 ) ↔ ∃ 𝑧𝑤𝑥𝐴 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
7 elcnv2 ( 𝑦 𝑥 ↔ ∃ 𝑧𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
8 7 rexbii ( ∃ 𝑥𝐴 𝑦 𝑥 ↔ ∃ 𝑥𝐴𝑧𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
9 rexcom4 ( ∃ 𝑥𝐴𝑧𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) ↔ ∃ 𝑧𝑥𝐴𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
10 rexcom4 ( ∃ 𝑥𝐴𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) ↔ ∃ 𝑤𝑥𝐴 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
11 10 exbii ( ∃ 𝑧𝑥𝐴𝑤 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) ↔ ∃ 𝑧𝑤𝑥𝐴 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) )
12 8 9 11 3bitrri ( ∃ 𝑧𝑤𝑥𝐴 ( 𝑦 = ⟨ 𝑧 , 𝑤 ⟩ ∧ ⟨ 𝑤 , 𝑧 ⟩ ∈ 𝑥 ) ↔ ∃ 𝑥𝐴 𝑦 𝑥 )
13 1 6 12 3bitri ( 𝑦 𝐴 ↔ ∃ 𝑥𝐴 𝑦 𝑥 )
14 eliun ( 𝑦 𝑥𝐴 𝑥 ↔ ∃ 𝑥𝐴 𝑦 𝑥 )
15 13 14 bitr4i ( 𝑦 𝐴𝑦 𝑥𝐴 𝑥 )
16 15 eqriv 𝐴 = 𝑥𝐴 𝑥